The Euro Team and the SWM Network

Principal components do not necessarily have an orientation. However, when you are making principal components from networks of tree ring widths, it’s a good idea to try to think about physical interpretations. Here’s a funny example where the Euro Hockey Team has lost its way. They observe the following:

for the proxy principal components in the MBH collection the sign is arbitrary: these series have, where necessary, had the sign reversed so that they have a positive correlation with the northern hemisphere temperature record).

Now there’s an interesting way to illustrate the potential pitfalls of this assumption in the Stahle/Southwestern US-Mexico network where the Euro Hockey Team has unwisely waded in. The EHT has made duplicate calculations in most AD1400 situations of PCs without using series extended to 1980. (As I noted in a post, in MM03, we noted that many of the series had been extended to 1980, but it’s not a point that we dwelled on and I’m pretty sure that we subsequently said that this was not an issue that we were particularly fussed about.) Nonetheless, the EHT have re-done all these calculations. Maybe they were checking to see, if by reducing the number of series being used, they could enhance the “desired signal” – this is perhaps the case in the NOAMER network. Trying to enhance the “desired signal” is an established dendrochronological procedure as stated by elsewhere by coauthor Esper as follows:

this does not mean that one could not improve a chronology by reducing the number of series used if the purpose of removing samples is to enhance a desired signal. The ability to pick and choose which samples to use is an advantage unique to dendroclimatology.

In doing so, they might not have noticed that the SWM network declines to only 2 series in the AD1400 network in their 00 network option. While they don’t say that they only used 2 series in this network, I’ve been able to replicate series #11 in archived mitrie_new_proxy_pcs_1400_v01.nc using two series obtained through this culling and am confident that this is what they did. Now the PC1 of two series is the average – which has a fairly obvious interpretation. In this particular case, I’m satisfied that the two series in the Euro Team “network” are the earlywood and latewood widths of Cerro Durango and Los Angeles Sawmill, WDCP/ITRDB codes mexi023e and mexi023l (which I’ve specifically compared and matched to Mann series swmxdfew09.dat and swmxdflw09.dat.

Figure 1 below shows the plot of the Euro SWM PC1 and the two contributing series. As you see, the underlying series have greater ring widths in the 15th century than in the 20th century. The site in question is at 3170 meters. We’re told that ring widths of these high-altitude sites are supposed to be linearly correlated to temperature so the ring widths themselves are supposed to have some physical meaning. However, in establishing an orientiation for the SWM PC1, the EHT have flipped them over “so that they have a positive correlation with the northern hemisphere temperature record”, disregarding the presumed physical interpretation of the ring widths. While this, in some small way, aids the project of enhancing 20th century values relative to 15th century values, the flipping of the PC1, in this particular case, seems a little, shall we say, opportunistic on the part of the Team.

Figure 1. PC1 is downloaded from mitrie_new_proxy_pcs_1400_v01.nc. The EW and LW series are Mann’s versions, which can be traced to mexi023e and mexi023l, Cerro Durango and Los Angeles Sawmill, PSME, 3170 m.

Does it “matter”? I don’t think that it “matters” very much to the final reconstruction. But, at this point, people should be trying for a little craftsmanship.

This little fiasco also nicely points out one of the problems with using PCs at all. PCs discard information on the orientation of a series – whether it points up or down, requiring later potentially ad hoc interpretations of the results using expedients like correlation with NH temperature to determine whether the PC series is pointing up or down. It would be better not to discard the information on the orientation. If the Euro Team were actually trying to advance the state of the art, that’s what they’d be thinking about. As it is, I think that someone on the Euro Team deserves a few minutes in the penalty box, don’t you?

49 Comments

Steve, you are right about this, too. If half the dendro records end up with negative signs (PCA assigns these automatically–really could not care less–but from a scientific POV it is hugely important), it makes me wonder about underlying physics. What do these records correspond to? Is there any “global” signal at all?

It is also disheartening that, apparently, the HT is not concerned about this.

Does this sort of thing really occur in other areas of science? For example, (and I realize this is a bit of a stretch), when a pharmaceutical company tests a new drug and finds that half the people taking the drug live 10 years longer and half live 10 years shorter, do they conclude that a) The drug has an enormous impact on life expectancy, and b) the drug increases life expectancy (assuming the first PCA happens to have received the right sign)?

In our red noise simulations of the Mannian “PC” as applied to the NOAMER network, of the series contributing strongly to the red noise PC1, half were upside-up and half were upside-down. The PC algorithm didn’t care; it just assigned signs to make the maximal HS. In the actual network, to do so, it picked out bristlecones which all had an upward orientation. But for the purposes of the PC algorithm, if half the bristlecone ring width chronologies had been inverted, the MAnnian (or any other PC algorithm) wouldn’t have cared. The information that the bristlecone chronologies were upward pointing was discarded in the chronology. In the MBH99 illustration, the series was flipped over as the PC1 pointed down.

Ritson went crazy about this. He didn’t get that the PC algorithm automatically assigned orientations and accused us of manually flipping the red noise series over in our red noise simulations, fulminating about Dan Rather and Rather-gate. I wonder whether he will comment about the Euro Hockey Team flipping series over.

This illustrates the danger of PCA on time-series. If you’re going to do it you had better be careful about your interpretations. Arbitrary flipping of signs to get a better fit to the instrumental record with little or no regard for what happens with the pre-instrumental part of the PC – that is negligent.

It’s pretty neat that they’ve inadvertently provided an example which illustrates the problem so exactly. It’s hard to imagine someone in ordinary circumstances doing PCA on a network with only 2 series in it – here it undoubtedly was unintentional. But it’s always a good idea to see what methodologies do in trivial cases.

If half the dendro records end up with negative signs (PCA assigns these automatically–really could not care less–but from a scientific POV it is hugely important), it makes me wonder about underlying physics. What do these records correspond to? Is there any “global” signal at all?

This is overly skeptical. The trees are a weak proxy for temperature, no dispute there from either team. Also, you can’t dispute that the signal, however weak, is global.

The real problem here is the interpretation given to the PCs. If the signing by eigenanalysis is arbitrary (and it is) then the issue here is how those series are re-signed. If they are re-signed so that all blades are pointing up, then you may be eroding the MWP.

How could this happen? If the interpretation of each PC is that it is a proxy for the instrumental record, then that is your first mistake. If the first PC is the HS with rising blade, some of those PCs with the blade pointing down will also be a (+) proxy for the instrumental record. “Hunh?!” you say. Think about it. Some regions of the world do not have a HS temperature signature. When a lesser PC has a down-pointed blade, it is an indication that it is deviating from the mean PC1 during the instrumental record in order to bring that region’s reconstructed temperature back down in line with reality. During the pre-instrumental era, however, it might actually correlate with PC1! If you turn the blade upside-down, you turn the handle upside down. Away goes your MWP.

The source of the problem is that although PCA gurantees orthogonal PCs, it does not guarantee orthogonality among smaller pieces of PC time-series. i.e. The blades of two PCs may be anticorrelated with shafts correlated, thus producing a net zero correlation.

That is why PCA on time-series is dangerous. Interpretation must be VERY cautious and methodical, not cavalier. Something that is hard to do using a global network of data that one has not collected oneself.

#7 This brings up something I don’t understand about using PCA in this context: although the generated components are orthogonal, the natural variables they are being mapped on to are not. The natural variables (temperature, humidity, etc) are not independent. This means that there cannot be a one-to-one correspondance between components and variables. Is there some technique for dealing with this?

Re #8
Remember this is spatiotemporal PCA. They are interested in reconstruction of (a) the global mean field of temperature over time and (b) regional deviations from that global mean temperature vector. It is possible, therefore, that the regional deviations are orthogonal to the mean, and also to each other. i.e. The tendency is to interpret components as independent regional temperature signals, not non-independent multi-variate climatic signals.

There is an answer to your question. But it would spawn a whole other side-discussion which would be largely off-topic.

Am hoping #7 draws a comment from Steve M. I’ve not looked at one of these PCAs before and after re-signing of the PCs by the Team. The result would be very telling. Have you done this Steve (or Ross M)?

Here’s my question. Is the re-signing done entirely on the basis of the 20th c. “blade” direction? If so, then in each case, what happens to the “handle”? And ultimately, what would be the result if the PCs were correctly (IMO) interpreted as regional deviations from the global mean vector (as opposed to varying regional flavors of the 20th c. instrumental trend upward)?

If these questions don’t make sense, it may be worth hashing out offline. (Experience with this blog tells me it’s unlikely I’ve identified anything new that you haven’t already fully worked out.)

#10. bender, in effect, the Mannian PC method sorts out red noise series by blade direction and assigning negligible weight ot unbladed series. This sorting effect is dramatically enhanded if even one synthetic HS series is inserted into the mix. This is simply another perspective on our original red noise point.

What happens to the handle in a red noise situation is that, after sorting on blades, the red noise in the handles cancels out so that the variance is much reduced i.e. that’s why it is a shaft.

Principal components is the ONLY way that you can do this. For exampe, if you pick the 10 most upward HS series out of 36 synthetic series with moderate persistence, you get a HS with a reinforced blade and attenuated shaft. MAnn’s PC is really just an automated method of cherrypicking.

If there is an actual signal in a network of VZ-type pseudoproxies (signal plus uniform white noise), PC methodology (covariance or correlation) will recover the signal acceptably. In a sufficiently “tame” network, even Mannian PC methods will recover a signal in the PC1. Any lower-order PCs in pseudoproxy networks tend to be oscillations – which is obvious since they have to be orthogonal to the “signal”. In tree ring networks though, you can have distinct patterns like the bristlecones, which remain as aan eesentially orthogonal signal to the other trees – and emerge in one form or another as the Mannian PC1, covariance PC4 or correlation PC2.

I understand this, but it doesn’t answer my question. When they are “sorting out the series by blade direction” they are sorting on the re-signed series, and not the unadultered PCA output (which have “arbitrary” sign), correct? Assuming this is the case, then I want to know what the analysis looks like under various re-signing (i.e. interpretation) schemes.

#12. bender – we’re probably talking at cross-purposes: I don’t mean to review issues that are elementary for you but I’m missing the issue here. So pardon me if this seems simplistic. Since there are two stages in MBH98 – (1) the tree ring principal components ; (2) then the partial least squares (“inverse”) regression on the network of proxies and tree ring PC series. The latter step is against a trend so that signs on the PC series are sorted out through regression against the trend. Is this what you’re wondering about? Otherwise I’d appreciate it if you’d try to clarify the issue that’s nagging you a little more as I’m missing the point.

for the proxy principal components in the MBH collection the sign is arbitrary: these series have, where necessary, had the sign reversed so that they have a positive correlation with the northern hemisphere temperature record

So is there any record of which PC’s had their sign changed? I would like to see a pairwise plot of the unadultered PC output vs. a plot of those that had their sign changed (in order that the blade went up).

…

You see what I’m getting at here? I’m saying the down-turned blade may already have a valid physical interpretation (as a regional correction for the wacky bcp PC1). Assuming the contrary, e.g. that the sign is arbitrary and changing the sign to match the 20th c. is valid, may be giving the wrong interpretation to the PC!

Based on your reply in #11, you are either missing my point completely, or my point is just not stated well enough that you recognize it as something you’ve already dealt with in your past work. (It’s more than likely that I’ve missed something altogether, but we should cover that possibility. I’ve got this funny feeling I may be tripping across something neither group has thought of yet. (Due to my own experience with spatiotemporal PCA.))

I’m not sure that it make sense to say that the sign was “changed” in most circumstances. If the regression coefficient was negative, that’s the sign that he got. To fully survey this, you have to look at esoteric “relationships” between the temperature PC11 and the Stahle/SWM PC8. What would it mean to say that one or both of the signs was “changed” – I don’t have a view.

In the case of the MBH99 PC1, Mann overtly switched the sign to make the HS point up – but there’s a plausible interpretation of this since the HS was dominated by bristlecones with negative coefficients.

However, the Euro team seems to have overtly flipped series to make them all point upward in the archive – something that Mann himself didn’t do. I’ve made notes on what’s in their PC directory and it’s pretty comic. They’ve tested variations on the Mannian algorithm e.g. a 125 year blade instead of a 79-year blade; things like that. I can’t imagine what Wegman’s view of these effusions will be. It was one thing to say this stuff pre-NAS and pre-Wegman, but they’ve just ignored these reports.

Re #17
Ok, so my point is irrelevant wrt Mann, but is relevant wrt Team Euro. (It’s the point that matters, not who suffers from it.)

Flipping PC1’s sign is ok, because it is arbitrary. Systematically flipping the others according to some arbitrary criterion is not ok. From my POV the act of re-signing the PCs represents an interpretation of the PCs. (That’s not their POV. Possibly because they do not understand what they are doing.) My point is I dispute this interpretation.

If I could see a plot of the PCs and a list of which ones were re-signed, I think it should be possible to provide a better alternative interpretation of the lower PCs, as regional deviations from the “bcp/global mean” signal. A more logical interpretation of the PCs might preclude the arbitrary re-signing and might bring back the MWP.

fFreddy, eigenvalues are always positive. Let’s boil this back from the language of matrices – an “eigenvector” is simply a weighting factor which can have both positive and negative values. A “principal component” is no more and no less than a type of index that is obtained from applying a weighting factor (with positive and negative factors). Most of you are used to indexes that consist only of positive weights e,g, a Dow-Jones Index, a Consumer Price Index. However, you can also have things that are contrasts between the industrials and the techs or something like.

Now that I think about it, there are some funny examples of flipping in the regression phase (which is where the rubber hits the road for this issue.) Some of the instrumental temperature series used as “proxies” have negative correlations to NH temperature in the calibration period and are assigned negative coefficients i.e. flipped over. A higher instrumental temperature in 3-4 out of 11 locations (Trondheim is one if I remember correctly) will lead to lower reconstructed values of the final reconstruction. It’s hard to get a better example than that.

Re #20
That’s a slightly different problem from the one I’m describing, but stemming from similar origins, and equally disturbing. Hopefully my posts here can be clarified to the point where they make sense to you. I think there’s something significant here, and your #20 only serves to amplify that feeling.

#19 has got the idea right, but the language all wrong. The eigenvectors are composed of a suite of values that have positive and negative numbers. In our case these would be one number for each year. A change in sign means flipping the sign of every number in the eigenvector. i.e. multiplying through by -1, which is effectively turning the hockey stick (or whatever shape that eigenvector has) upside-down. Flipping the first eigenvector is generally ok because it’s interpretation as a global mean vector is usually unambiguous. Flipping the others *is* dispensing with information contained in the eigenvectors, for reasons outlined in earlier comments.

That’s exactly why I can’t believe that that’s what they do. If they are flipping as necessary so that the 20th c. trend points upward in every case, under the assumption that each PC is a surrogate for a unique mode of 20th c. temperature increase, then that is what I dispute. If you do this you will systematically melt away positive and negative anomalies prior to the 20th c. i.e. This would not only sharpen the blade of th HS, it would also straighten & flatten the shaft. The alternative interpretation of these down-pointing blades (and remember, these have a lot less eigenvalue weight to them) is as regional corrections to PC1, which, rather than being interpreted as a global temperature field, could be better be interpreted as a uniquely regional bcp signal.

Note how interpretation of PCs 2-8 is relative to your interpretation of PC1. (The orthogonality of the decomposition could arise from independent effects. Or it could arise from spatial independence. Or proxy uniqueness. On the other hand, the orthogonality might be highly superficial, with long sections of eigenvectors (e.g. during the MWP) being non-orthogonal. Flip them ALL over and you’ve lost your MWP.)

Steve M, I don’t want to say anything here that is incorrect or confusing to your audience. So you decide when you’ve heard enough. Just make sure not to gloss over these comments.

Re #22
I think I see your point: the bcps’s are so dominant in these reconstructions that the effects I’m describing, while potentially important in a normal reconstruction, make no substantive contribution to the final shape of the stick whatsoever. Absence of MWP, steepness of 20th c. rise – it’s almost all in the bcps.

If that’s true – and I think you may be right – then why do they do it? Why do they invert some of the PCs?

Well, someone here a while back pointed out that in essence all the proxies are pinned (to use the term we’ve been using lately) at two points. These are the two ends of the surface temperature record which is used to calibrate the proxies. So any proxy which doesn’t have a hockeystick blade won’t be given much of a weighing. And if it happened to have an inverted blade it may be flipped (assuming the process being used allows it) to give a better fit. Or, to say the same thing, it will have a negative weighing. All the calibration, of course, will be relative to the mean of the calibration period.

But since we’re dealing with a calibration period with a rising trend, the bottom pin has to be below the mean so proxies which dip down before rising in the 20th century are weighed higher than those which don’t dip before rising, creating the handle, quite irrespective of the LIA. Likewise, since there’s no real constraint on what a proxy can do as long as it behaves itself in the calibration period, they will on average return to the mean so that any warmer than average period will be prominent, quite irrespective of the MWP. But in general any real signal for either LIA or MWP in the final reconstruction will be muted since a proxy which doesn’t have a real temperature signal but accidentally at least partially matches the calibration signal will be weighed higher and is as likely as not to cancel out the actual temperature signals outside the calibration period.

I’m terribly sorry, but I think that’s not right (or else we are talking about different things). In the language of matrices, eigenvalues can be negative – it means that the effect of the matrix is to flip the eigenvector into the opposite direction. You can’t just change the sign on both the eigenvector and its eigenvalue, and say the eigenvalue is now positive; the “negative eigenvector” needs to get flipped as well.

there are some funny examples of flipping in the regression phase (which is where the rubber hits the road for this issue.)

Hence my comment, because this is where the eigenvalues get thrown away.
Surely if you have a PC1 with 38% of explained variance and a PC5 with 8%, and you are treating them as equivalent, then you are inflating the significance of the PC5 by 38/8 = 4.75 times ?
So what is the effect of ignoring such a fundamental thing as the sign of the eigenvalue ?

Actually, the answer is probably that I should stop swanning around making airy-fairy comments and get stuck in a bit more …

Re #24
I appreciate the effort, but this is sufficiently vague that it is unhelpful. Nothing here I did not know already and it does not address the issue of concern, which is, again:

for the proxy principal components in the MBH collection the sign is arbitrary: these series have, where necessary, had the sign reversed so that they have a positive correlation with the northern hemisphere temperature record

Why are the PC signs described as “arbitrary” and why are they being selectively reversed, especially given #22 – that the bcps are running the show, such that sign reversal should have no effect on the stick’s shape.

#25 For the case at hand, positive-semi-definite matrices, SteveM is correct in asserting that eigenvalues are always positive (or, with probability 0, 0). In fact, the opposite result is also true: If all eigenvalues are .ge. 0, then the matrix is positive-semi-definite.

Obviously, for the general case, which is not relevant here, you are right that eigenvalues can be negative.

PCA is eigenanalysis of the proxy covariance (or correlation) matrix. I imagine that is going to put some constraint on the eigenvalues.

But on the subject or arbitrariness.

The help menu from R’s princomp (PCA) algorithm states:

The signs of the columns of the loadings and scores are arbitrary, and so may differ between different programs for PCA, and even between different builds of R.

Three questions, then:
(1) Does the fact that the signs are arbitrary mean that they are going to be arbitrary when eigenvalues are strictly ge.0?
(2) Does the fact that the signs are arbitrary mean that the eigenvectors can be re-signed? Would that not destroy the eigenspace of the decomposition?
(3) Does the fact that the signs are arbitrary mean that one can change some of the eigenvectors without worrying about others? (What exactly is it that is arbitrary – the individual signs, or the entire vector of signs?)

#30 (3) – NO, NO, NO. The only thing arbitrary is whether the vector is multiplied by +1 or -1. I recall just enough group theory from university that I sometimes think of PCs as being like a coset of 2 elements – the +1 version and the -1 version, which thus are an unoriented element. IN a way, it’s like a line in 3-space. You can specify the line with two different direction vectors – one pointing each way up and down the line. But the line itself doesn’t have an orientation.

The only thing arbitrary is whether the vector is multiplied by +1 or -1

Yes, I know, and that’s what I’m talking about. I’m asking for clarification as to the implications of this arbitrariness. If each vector is thought of as having a sign, then can you freely re-sign any of the vectors, or are you constrained in some way?

Now there’s an actually an interesting point in these eigenvalues arising from fFreddy’s point that I hadn’t thought about before. I don’t think that it matters a speck in terms of reconstructions but it’s an interesting theoretical point. A principal components analysis operates on positive semidefinite matrices (covariance or correlation) and thus has positive eigenvalues. However Mann didn’t do a principal components analysis; he did a singular value decomposition on the de-centered data matrices. However you still seem to get positive eigenvalues – so there must be weaker conditions than positive semidefinite which still generate positive eigenvalues.

Where do they get off with all this? bender, here are some old notes of mine on Cook et alal 1994, a survey of multivariate methods by key Hockey Team authors including Jones and Briffa.

It begins by a consideration of OLS as applied to short calibration periods where the practitioner has large matrices of both predictors and predictands. Here some care needs to be taken in not confounding predictor-predictand with cause-and-effect. In Cook et al [1994], the predictors X are data sets of tree ring width measurements (“tree ring networks”) while the predictands Y are climate gridcell series. Here the predictand is the cause rather than the effect. The regression of cause against effect is an inverse regression as understood in English-language statistical literature [Lucy et al 2004], rather than a classical regression where the effect is regressed against the cause. (Von Storch et al 2004 and BàÆà⻲ger et al 2006 appear to use the term inverse regression in a different sense and we urge that readers exercise some caution in ensuring that they have familiarized themselves with the specific methods of these two articles to ensure that they draw appropriate conclusions from them.)

Applying the viewpoint of Cook et al 1994 to the MBH98 network of 415 proxy series and 1082 gridcells, mechanical application of OLS methods as presented by Cook et al – see their equation (2) – would lead to the generation of 449,030 coefficients from only 32,785 measurements. Cook et al. observe:

“experience in reconstructing climate from tree rings indicates that such models frequently produce reconstructions that cannot be verified successfully when compared with climate data not used in estimating the regression coefficients. This can happen regardless of the statistical significance of the overall regression equation.”

The last statement comes as no surprise to a third party.

Cook et al. then note the possibility of “best subset regression”, noting that best-subset methods for the “simple case of 1 predictand” are well known, referring to variants of stepwise regression, all possible regressions, Mallow’s Cp and Allen’s PRESS, citing Draper and Smith 1981. They note the potential for “artificial predictability” due to a posteriori selection of the true predictors.

bender, Mann’s PC methods need to be viewed in the context of Cook et al. In his AD1820 calibration, he generates 16*112 regression coefficients from 79*112 measurements, from which 11*112 coefficients are selected.

bender, think it this way: you have a (symmetric) positive semidefinite matrix A. Then you can make the spectral decomposition A=UDU^T, where U is orthogonal (eigenvectors) and D is diagonal (with non-negative values, eigenvalues). Now take an arbitrary diagonal matrix L with +/-1’s in the diagonal. Then you can also write A=(UL)D(UL)^T and (UL) is still an orthogonal matrix. In the complex case, the indeterminancy is e^(ia).

Of course orthogonality is maintained under any diagonal L with +/-1’s on the diagonal. I’m not asking about orthogonality.

But while I have you here on orthogonality, Jean S, please consider this point, edited from #7:

although PCA guarantees orthogonal PCs, it does not guarantee orthogonality among smaller pieces of PC time-series. i.e. The blades of two PCs may be uncorrelated, but with shafts correlated, thus producing a net zero correlation

Changing a sign does not affect orthogonality of the eigenspace. But it will affect the correlation structure among pieces of the eigenvectors.

Sorry for being dense, guys, but I’m having a hard time following what you are writing. And I don’t think you are seeing my point. I could illustrate it if you posted up, as requested in #16, a plot of the PCs prior to any flipping. (Url to CA link would be fine.)

Yes, if you calculate the correlation among the parts of series (eg. instru against eigenvector), then changing the sign of the eigenvector changes the sign of that correlation. Whether that’s problematic, depends if you want to give physical meaning to PCs. I usually do not want to do that, but prefer to thing that retained PCs are some (possibly linear) combination of (scaled) underlying true signals. So in my thinking, the use of PCA is just to compress data and to get rid of some noise, not to actually extract (exact) information. In the case of tree rings, I think after a (proper) PCA you should still have all information affecting trees (temperature, precipitation…) in the retained PCs. If you want actually to extract information, then one should use Independent Component Analysis or similar methods.

#42. Orthogonality on one time scale does not imply orthogonality on other time scales, but they can often be “near orthogonal”.

In fact, there’s an interesting botch of this in MBH. He does PCA on the temperature grid cells by month, stating incorrectly that months were necessary because you need more time periods than gridcells to do PCA. Nature referees were unequal to the task of identifying this faux pas. After calculating monthly PCs, he then annualized them. The resulting series are no longer orthonormal; they are close to being orthogonal but depart from orthogonality.

Then he does one more thing – the annual series were calculated over 1902-1993, but he then truncates to 1902-1980 for matching to proxies, further moving the series from orthogonality.

Wahl and Ammann calculate temperature PCs annually maintaining orthogonality at this stage, but then truncate like Mann, so, once again, they are not orthonormal or orthogonal, thought they are close to it.

In the 15th century step, the proxies which are supposed to have a "signal" are surprisingly close to being orthogonal. So the PLS regression ends up being very similar to an OLS regression. There has really been far too little attention paid to the regression aspects: think about it- he is regressing a series of length 79 against 22 near-orthogonal proxies.

Where it gets really funny is in the no-PC case of Wahl and Ammann. Then they end up doing a PLS regression of a series of length 79 against 95 proxies.

so there must be weaker conditions than positive semidefinite which still generate positive eigenvalues.

Interesting point. For a symmetric matrix M, eigenvalues are positive if and only if the M is positive semidefinite. For a non-symmetric matrix, eigenvalues are positive if the symmetric matrix 0.5*(M+M’) is positive semidefinite.

I’m not sure what theorems might apply to the Mannomatic. Is it possible we’re looking at a new branch of linear algebra? ;-)

#46. Not a chance. Mathematicians would have been all over this for generations. I still have my linear algebra text: Greub- Linear Algebra, 3rd ed 1967. (Greub was my prof.) I’ll see what it says. It must have been hot off the press as I took this course in my 3rd year of university 1967-68.

I was trying to point out, clumsily it seems, that the HT (domestic U.S. and now Euro) has asserted the right to invent, and employ prior to testing, extraordinary statistical methods, so perhaps we should expect no better when it comes to linear algebra.

#48 Steve, I have been playing with the ArDec package for about an hour now, and I’m really not sure what it does. The package documentation suggests it might be useful, but I must be missing something when it comes to applying it. For example, plot(ts(data=1:468,start=c(1959,1),end=c(1997,12),frequency=12),ardec.trend(co2)$trend) yields a less convincing trend plot than what one gets from the raw data (plot(ts(data=1:468,start=c(1959,1),end=c(1997,12),frequency=12),co2). I’ll take a look at West [1997] when I get to work — maybe that will explain everying.